Lamb shift determination

The Zemach correction is essentially a nontrivial weighted integral of the product of electric and magnetic densities, normalized to unity. It cannot be measured directly, like the rms proton charge radius which determines the main proton size correction to the Lamb shift (compare the case of the proton size correction to the Lamb shift of order Za) in (6.13) which depends on the third Zemach moment). This means that the correction in (11.4) may only conditionally be called the proton size contribution. 

Both the theoretical and experimental data for the classic 2S i/2 — 2Pi/2 Lamb shift are collected in Table 12.2. Theoretical results for the energy shifts in this Table contain errors in the parenthesis where the first error is determined by the yet uncalculated contributions to the Lamb shift, discussed above, and the second reflects the experimental uncertainty in the measurement of the proton rms charge radius. We see that the uncertainty of the proton rms radius is the largest source of error in the theoretical prediction of the classical Lamb shift. An immediate conclusion from the data in Table 12.2 is that the value of the proton radius [27] recently derived form the analysis of the world data on the electron-proton scattering seems compatible with the experimental data on the Lamb shift, while the values of the rms proton radius popular earlier [28, 29] are clearly too small to accommodate the experimental data on the Lamb shift. Unfortunately, these experimental results are rather widely scattered and have rather large experimental errors. Their internal consistency leaves much to be desired. 

The intervals of gross structure are mainly determined by the Rydberg constant, and the same transition frequencies should be used both for measurement of the Rydberg constant and for measurement of the IS Lamb shift. 

To obtain unbiased results we have calculated self-consistent values of the IS Lamb shift which are collected in last seven lines of Table 12.3. These values being formally consistent are rather widely scattered. Respective self-consistent values of the classic Lamb shift obtained from the experimental data in [31, 32, 33, 34, 35, 36] are presented in Table 12.2. The uncertainty of the self-consistent Lamb shifts is determined by the uncertainties of experimentally measured frequencies used for their determination. Typically there are two such frequencies. One is usually fis-2s, and it is now measured with a very high accuracy of 1.8 parts in 10 [36]. The other frequency is measured less precisely and its experimental uncertainty determines the uncertainty of the self-consistent Lamb shifts. There are good experimental perspectives for measuring the second frequency with higher accuracy [33, 36]. 

Determination of the most precise value of the Rydberg constant requires a comprehensive analysis of results of the same experiments used for determination of the 15 Lamb shift [26, 31, 32, 33, 34, 35, 36]. This analysis was performed in [1], and resulted in the value in (12.1) which has relative uncertainty 6 = 6.6 X 10. This uncertainty is limited by the experimental... 

The main contribution to the hydrogen-deuterium isotope shift is a pure mass effect and is determined by the term E in (3.6). Other contributions coincide with the respective contributions to the Lamb shifts in Tables 3.2, 3.3, 3.7, 3.9, 4.1, 5.1, and 6.1. Deuteron specific corrections discussed in Subsubsect. 6 and collected in (6.16), (6.28), (6.29), and (6.37) also should be included in the theoretical expression for the isotope shift. 

All yet uncalculated nonrecoil corrections to the Lamb shift almost cancel in the formula for the isotope shift, which is thus much more accurate than the theoretical expressions for the Lamb shifts. Theoretical uncertainty of the isotope shift is mainly determined by the unknown single logarithmic and nonlogarithmic contributions of order ZaY m/M) and a(Za) (m/M) (see Sects. 4.3 and 5.2), and also by the uncertainties of the deuteron size and structure contributions discussed in Chap. 6. Overall theoretical uncertainty of all contributions to the isotope shift, besides the leading proton and deuteron size corrections does not exceed 0.8 kHz. 

The natural linewidth of the 2P states in muonic hydrogen and respectively of the 2P — 2S transition is determined by the linewidth of the 2P — IS transition, which is equal hP = 0.077 meV. It is planned [64] to measure 2P — 2S Lamb shift with an accuracy at the level of 10% of the natural linewidth, or with an error about 0.008 meV, which means measuring the 2P — 2S transition with relative error about 4 x 10 . 

P — 2S Lamb shift in hydrogen will be reduced to a comparable level, it would be possible to determine the proton radius with relative error smaller that 3 X 10 or with absolute error about 2 x 10 fm, to be compared with the current accuracy of the proton radius measurements producing the results with error on the scale of 0.01 fm. 

This experimental development was matched by rapid theoretical progress, and the comparison and interplay between theory and experiment has been important in the field of metrology, leading to higher precision in the determination of the fundamental constants. We feel that now is a good time to review modern bound state theory. The theory of hydrogenic bound states is widely described in the literature. The basics of nonrelativistic theory are contained in any textbook on quantum mechanics, and the relativistic Dirac equation and the Lamb shift are discussed in any textbook on quantum electrodynamics and quantum field theory. An excellent source for the early results is the classic book by Bethe and Salpeter [6]. A number of excellent reviews contain more recent theoretical results, and a representative, but far from exhaustive, list of these reviews includes [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. 

Hereafter we put /ig = 1. Below we express our results in terms of the statistical properties (correlators) of the environment s noise, X(t). Depending on the physical situation at hand, one can choose to model the environment via a bath of harmonic oscillators [6, 3]. In this case the generalized coordinate of the reservoir is defined as X = ]T)Awhere xi are the coordinate operators of the oscillators and Aj are the respective couplings. Eq. 2 is then referred to as the spin-boson Hamiltonian [8]. Another example of a reservoir could be a spin bath [11] 5. However, in our analysis below we do not specify the type of the environment. We will only assume that the reservoir gives rise to markovian evolution on the time scales of interest. More specifically, the evolution is markovian at time scales longer than a certain characteristic time rc, determined by the environment 6. We assume that rc is shorter than the dissipative time scales introduced by the environment, such as the dephasing or relaxation times and the inverse Lamb shift (the scale of the shortest of which we denote as Tdiss, tc [Pg.14]

P. Indelicato, S. Boucard, E. Lindroth, Relativistic and many-body effects in K, L, and M shell ionization energy for elements with 10 < Z < 100 and the determination of the Is Lamb shift for heavy elements, Eur. Phys. J. D 3 (1998) 29. 

Before the individual parts of this function are discussed, the energy eigenvalue will be considered. The ground state energy g of the helium atom is just the energy value for double-ionization which can be determined accurately by several different kinds of experiments. Before the experimental value can be compared with the calculated one, some small corrections (for the reduced mass effect, mass polarization, relativistic effects, Lamb shift) are necessary which, for simplicity, are... 

The absolute frequency of the fundamental IS — 2S transition in atomic hydrogen has now been measured to 1.8 parts in 1014, an improvement by a factor of 104 in the past twelve years. This improvement was made possible by a revolutionary new approach to optical frequency metrology with the regularly spaced frequency comb of a mode locked femto-second multiple pulsed laser broadened in a non-linear optical fiber. Optical frequency measurement and coherent mixing experiments have now superseded microwave determination of the 2S Lamb shift and have led to improved values of the fundamental constants, tests of the time variation of the fine structure constant, tests of cosmological variability of the electron-to-proton mass ratio and tests of QED by measurement of g — 2 for the electron and muon. 

One can study muonic atoms [14,15,16]. The muon orbit lies lower and much more close to the nucleus and its energy levels are much more affected by the strong interactions. However, to determine the nuclear contributions (for e. g. the one for the Lamb shift, which is completely determined by the nuclear charge radius) it is not necessary to know the QED part with an accuracy as high as in the case of the hydrogen atom. As a result, one can try to determine the parameters due to the nuclear structure and apply them afterwards to normal atoms. 

By comparison of one quarter of the IS1 — 25 transition frequency with the 25 — 45 and 25 — 4D transition frequency, the main energy contributions described by the simple Rydberg formula are eliminated. The remaining difference frequency (about 5 GHz) is determined by well known relativistic contributions, the hyperfine interaction, and a combination of Lamb shifts. Since quantum electrodynamic contributions scale roughly as 1/n3 with the principal quantum number, the Lamb shift of the 15 level is the largest. 

A small part of the infrared light was frequency doubled in a KNbC>3 crystal to 486 nm and combined with the blue light of the dye laser that excites the IS —2S transition. A fast photodiode is used to observe the frequency difference v(2S — AS/AD) — 1/4 v(lS — 2S). Fitting the 2S —4S and 2S — AD line profiles with a theoretical model calculated by Garreau et al. [17,18,19] and correcting for some systematic effects, the ground state Lamb shift could be determined with an accuracy of 1.3 parts in 105, one order of magnitude more precise than in previous measurements [21,22]. 

As already mentioned in the second part of this review, we made an average of these different determinations of R00 by performing a least squares adjustment [72] which takes into account all the precise experiments the measurements of the 251/2 Lamb shift, the optical frequency measurements of the 15— 25 and 25 — nD transitions in hydrogen and deuterium, and also the measurements of... 

In order to extract the QED or nuclear effects from the 1S-2S frequency, a second frequency must be known. The present uncertainty in the Lamb shift and Rydberg constant is determined by the accuracy of such a measurement. The most precise measurements have been made on transitions from 2S to higher levels in a super-thermal beam of metastable 2S atoms [19]. As will be described, ultracold hydrogen offers possibilities for significant improvements. 

The spectrum of hydrogen is composed of a major structure, determined by the Rydberg constant, QED corrections like the Lamb shift and finally nuclear shape... 

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